multiphysics model development to characterize ......mechanical responses of these hydrogels remain...
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This document is downloaded from DR‑NTU (https://dr.ntu.edu.sg)Nanyang Technological University, Singapore.
Multiphysics model development to characterizefundamental mechanism of bio‑responsivehydrogels
Goh, Kek Boon
2019
Goh, K. B. (2019). Multiphysics model development to characterize fundamentalmechanism of bio‑responsive hydrogels. Doctoral thesis, Nanyang TechnologicalUniversity, Singapore.
https://hdl.handle.net/10356/81279
https://doi.org/10.32657/10220/47498
Downloaded on 22 Jun 2021 06:54:30 SGT
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MULTIPHYSICS MODEL DEVELOPMENT
TO CHARACTERIZE FUNDAMENTAL
MECHANISM OF BIO-RESPONSIVE
HYDROGELS
GOH KEK BOON
School of Mechanical & Aerospace Engineering
A thesis submitted to the Nanyang Technological University
in partial fulfillment of the requirements for the degree of
Doctor of Philosophy
2019
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Acknowledgement
I like to express my gratitude to my supervisor Professor Lam Khin Yong and
co-supervisor Associate Professor Li Hua for guiding me over the years, and
also my thesis advisory committee (TAC), Dr Cui Fangsen and Associate
Professor Li Lin, for their suggestions during my Ph.D. program. Their push to
pursue good and many works will stay with me even after my Ph.D training.
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Abstract
Environmentally bio-responsive hydrogel is of tremendous interest in plethora
of advanced engineering applications, such as hydrogel-based urease-loaded
dialysis membranes and hydrogel-based hemoglobin-mediated oxygen carrier.
However, literature reviews reveal that the coupled bio-chemo-electro-
mechanical responses of these hydrogels remain poorly understood, due to lack
of accurate mathematical models to numerically characterize the
environmental-induced hydrogel synergistic performances. Therefore, the
present research work focuses on the development of multiphysics models to
obtain a deeper understanding of such materials, especially when operating at
extreme of physiological environmental conditions.
The first academic achievement made in this present work is the
development of a multiphysics model for describing the coupled bio-chemo-
electro-mechanical responses of urease-loaded hydrogel. By coupling the
multiphysics interactions occurring in the hydrogel together, the model consists
of three governing mass, momentum and energy conservation equations, and
also four sets of constitutive relations, namely mass flux, fixed charge equation,
and nonlinear mechanical equation. A novel rate of reaction is also incorporated
into the model to describe the urease activity as a function of ambient
temperature coupled with environmental solution pH, capturing the
environmental-sensitive urease ionization and denaturation states. For model
validation, the multiphysics model is examined with the comparison between
present numerical results and published experimental observations, where good
agreements are achieved, especially for temperature-, pH- and urease-induced
swelling deformations and urease catalytic activity of the polyelectrolyte
hydrogels. The result shows that the urease catalytic activity patterns differ in
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anionic and cationic urease-loaded hydrogels by increasing the environmental
concentration of sodium chloride at a relatively higher environmental
concentration of urea, whereas the urease catalytic activity remains almost
unchanged when the environmental pH increases above the acid-base
dissociation constant pKa of the polyacidic hydrogel. The result also shows that
the osmotic pressure response of urease-loaded hydrogel enlarges linearly by
increasing physiological urea concentration making it biocompatible for
healthcare diagnostic applications.
The second academic achievement is the development of another
multiphysics model to elucidate the coupled-stimulated responses of
hemoglobin-loaded polyelectrolyte hydrogels. A developed constitutive relation
is integrated into the model to capture immobile hemoglobin bioactivity as a
function of ambient oxygen coupled with environmental pH. After validation
against the reported experimental observations, it is taken that the multiphysics
model can effectively characterize the hemoglobin saturation with oxygen for
(1) neonatal and (2) adult hemoglobins, and also the pH-induced swelling
deformation of hemoglobin-loaded polyelectrolyte hydrogels. The result
demonstrates that the hydration-induced swelling deformation of the
polyampholytic hydrogel changes in a bowl-shaped fashion by increasing the
environmental pH value, in which the pH-induced swelling deformation of
initially balanced polyampholytic hydrogel changes from a “bowl” to “V”-
shaped like pattern with decrease of immobile acidic and basic components
ionization strength. In addition, the result demonstrates that the strength
increase of both the immobile acidic and basic components in the initially
balanced polyampholytic hydrogel causes the hydrogel to exhibit isoelectric
point behavior at wider environmental pH range, whereas the initially
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unbalanced polyampholytic hydrogel collapses at the environmental pH
coinciding with dissociation constant of the dominant immobile charge group,
if the initial dominant immobile charge group concentration is twice that of the
counter one.
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Table of Contents
Acknowledgement …………………………………………………………..iii
Abstract……………………………………………………………………….v
Table of Contents……………………………………………………………viii
List of Figures…………………………………..………………………….…xii
List of Tables……………………….………………………………………xviii
Chapter 1: Introduction……………………………………………………..18
1.1 Background and Motivation……………………………………..18
1.2 Objective and Scope……………………………………………..20
1.3 Outline …………………………………………………………..21
Chapter 2 Literature Review………………………………………………...23
2.1 Experimental Investigation on Bio-responsive Hydrogels……...23
2.1.1 Urease Immobilized Hydrogel………………………………26
2.1.2 Hemoglobin-loaded Hydrogel……………………………….29
2.1.3 Other Bio-responsive Hydrogels…………………………….32
2.2 Modeling the Equilibrium of Hydrogel Responses and Protein
Activities………………………………………………………...33
2.2.1 Transport Model……………………………………………..33
2.2.2 Multiphasic Mixture Model………………………………….36
2.2.3 Urease Enzymatic Activity…………………………………..37
2.2.4 Reaction Of Hemoglobin With Ligands……………………..39
2.3 Remarks………………………………………………………….41
Chapter 3 Multiphysics Model Development to Investigate the Reactive
Behaviors of Urease-loaded Hydrogel………………………….43
3.1 Introduction………………………………………………………..43
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3.2 Model Formulation………………………………………………...44
3.2.1 Conservation of Mass………………………………………..45
3.2.2 Conservation of Momentum…………………………………45
3.2.3 Free Energy Imbalance Inequality…………………………..45
3.2.4 Biochemistry for Constitutive Equation……………………..48
3.2.5 Electrical Field for Constitutive Equation…………………...50
3.2.6 Mechanical Field for Constitutive Equation………………...50
3.2.7 Boundary Conditions………………………………………...52
3.3 Validation of Model………………………………………………..53
3.3.1 Mechanical Behaviors of the Hydrogel……………………...53
3.3.2 Biochemical Activities of the Hydrogel……………………..57
3.4 Results and Discussion…………………………………………….61
3.4.1 How Does the Immobilized Urease Concentration Influence
Reactive Performances of the Hydrogel?..............................62
3.4.2 How Does Different Polymer Monomers used to Construct
Urease-loaded Hydrogel Influence its Urea-sensitive
Responses?............................................................................63
3.4.3 Osmotic Pressure of the Urease-loaded Hydrogel…………..62
3.4.3.1 Impacts of Environmental
Conditions…………………………………………………. 62
3.4.3.2 Impacts of Urease Kinetic
Properties………………………………………………… ..63
3.4.3.3Impacts of Initial Fixed Charge Group Density
……………………………………………………..64
3.4.4 Urease Catalytic Behavior…………………………………...65
3.4.4.1 Influences of Urea Concentration…………………..66
3.4.4.2 Influences of Salt Concentration…………………...71
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3.5 Remarks……………………………………………………………80
Chapter 4 Development of a Multiphysics Model to Examine the
Stimulated Responses of Hemoglobin-loaded Polyelectrolyte
Hydrogels………………………………………………………...87
4.1 Introduction………………………………………………………..87
4.2 Development of Model…………………………………………….87
4.2.1 Hemoglobin Reaction Mechanism…………………………..89
4.2.2 Acid-Base Reaction Mechanism…………………………….90
4.2.3 Formulation of the Multiphysics Model……………………..99
4.3 Numerical Implementation……………………………… …… ...93
4.4 Model Validation with Experimental Observations……………….95
4.4.1 Saturation of Hemoglobin with Oxygen…………………….95
4.4.2 Hydration-induced Deformation Responses of the Hydrogel.96
4.5 Results and Discussions……………………………………………98
4.5.1 Effects of Hemoglobin and Immobile Charge Group
Loading..................................................................................98
4.5.2 Effects of Ambient Oxygen Level…………………………...99
4.5.3 Effects of Environmental pH……………………………….110
4.5.4 Effects of Environmental Salt- and Oxygen-coupled
Stimuli……………………………………………………..110
4.6 Remarks…………………………………………………………..130
Chapter 5 Conclusions and Recommendations…………………………..143
Appendix…………………………………………………………………….136
References…………………………………………………………………...148
Publications Associated to this Thesis……………………………………..153
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List of Figures
Fig. 3.1 A general scheme for the systematic investigation of the urease-loaded
charged hydrogel-based system.
Fig. 3.2 Computational diagram of the multiphysics model to characterize the
system when subjected to environmental stimuli.
Fig. 3.3 Observation of swelling deformation for the hydrogel between the
numerical results by the model and Ogawa and Kokufuta’s
correlation, and their experimental data at initial radius 145.0 m .
Fig. 3.4 Observation of swelling deformation for the hydrogel between the
numerical results by the present multiphysics model and the
experimental data at different initial radiuses: (a) 235.5 m and (b)
318.5 m .
Fig. 3.5 Validation of the multiphysics model, capturing the urea-responsive
characteristics subject to environmental conditions: (a) The pH-
induced responsive behavior of the urea-sensitive hydrogel subject to
environmental pH (b) Thermally reactive performance of the
hydrogel from environmental 15 to 45.
Fig. 3.6 (a) The comparison between the published experimental observation
and the present numerical result for characterizing the pH-induced
urease catalytic activity in cationic hydrogel. (b) The hydration-
driven swelling performance of the hydrogel under varying
environmental pH.
Fig. 3.7 (a) The comparison between the published experimental observation
and the present numerical result for the pH-induced urease catalytic
activity in anionic hydrogel. (b) The pH of the hydrogel as a function
of the environmental pH.
Fig. 3.8 (a) The comparison between the published experimental observation
and the present numerically simulating result for the temperature-
induced urease catalytic activity in anionic hydrogel. (b) The pH of
the hydrogel as a function of the environmental temperature. (c) The
comparison between the published experimental observation and the
present numerical result for the temperature-induced swelling
hydration deformation of the urease-loaded hydrogel. (d) The
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contractive hydrostatic pressure acting on the hydrogel as a function
of the environmental temperature.
Fig. 3.9 (a) The activity of the immobilized urease and (b) the swelling ratio of
urease-loaded hydrogel as a function of urease concentration in the
hydrogel, when subjected to urea concentration of 30 mM with
environmental pH=7.0 and temperature 37 oC.
Fig. 3.10 (a) The urea-actuated hydration-induced swelling deformation of the
hydrogel as a function of concentration of urease, when subjected to
urea concentration 30 mM; and (b) The urea-actuated osmotic
pressure response p of the hydrogel as a function of concentration
of urease, when subjected to urea concentration 30 mM, with
environmental pH=7.0 and room temperature. The osmotic pressure
response p of the hydrogel is the differential osmotic pressure
between the urease-loaded hydrogel and its bare counterpart
( ) ( )0E Ep p C p C = − = .
Fig. 3.11 The osmotic pressure response p of urease-loaded hydrogel versus
concentration of urea ureaC for PHEMA, PNIPAM and PAAM
hydrogels, with environmental pH=7.0 and room temperature. The
osmotic pressure response p is the differential osmotic pressure
between the hydrogel immersed in urea-filled solution and the
hydrogel immersed in reference solution with ureaC = 0.
Fig. 3.12 The osmotic pressure response p subjected to variation
concentrations of urea ureaC , with environmental pH=7.0 and
temperature 37 oC at: (a) different environmental temperatures and
(b) variation of environmental pH. The osmotic pressure response p
is the differential osmotic pressure between the hydrogel immersed in
urea-filled solution and the hydrogel immersed in reference solution
with ureaC
= 0.
Fig. 3.13 The osmotic pressure response p subjected to variation
concentrations of urea ureaC , with environmental pH=7.0 and
temperature 37 oC for: (a) different urease catalytic constants kcat and
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(b) variation urease Michaelis constants KM, such as 0.26, 22.1 and
27.0 mM. The osmotic pressure response p is the differential
osmotic pressure between the hydrogel immersed in urea-filled
solution and the hydrogel immersed in reference solution with ureaC
= 0.
Fig. 3.14 The osmotic pressure response p of hydrogels loaded with urease
versus concentration of urea ureaC for different concentrations of
initial immobile charge group within the hydrogel, with
environmental pH=7.0 and temperature 37 oC. The osmotic pressure
response p is the differential osmotic pressure between the hydrogel
immersed in urea-filled solution and the hydrogel immersed in
reference solution with ureaC = 0.
Fig. 3.15 The urease activity of the anionic charged hydrogel as a function of
urea concentration ureaC
at different environmental: (a) temperature
and (b) pH.
Fig. 3.16 (a) The equilibrated responsive hydration and (b) its corresponding
urease activity of the anionic charged polymeric system as a function
of urea concentration ureaC
at different urease catalytic constants
kcat..
Fig. 3.17 (a) The equilibrated responsive hydration of the anionic charged
polymeric system as a function of urea concentration ureaC
at
different Michaelis constants KM and (b) the urease activity in the
charged hydrogel as a function of Michaelis constant KM at ureaC =
20 and 200 mM, respectively.
Fig. 3.18 (a) The equilibrated responsive hydration and (b) its corresponding
urease activity of the anionic charged polymeric system as a function
of urea concentration ureaC at different urease concentrations.
Fig. 3.19 The swelling deformation of the urea-sensitive charged hydrogel as a
function of environmental sodium chloride concentration NaClC at
different ureaC values for: (a) cationic and (b) anionic urease-loaded
hydrogels.
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Fig. 3.20 The urease activity of the charged hydrogel as a function of
environmental sodium chloride concentration NaClC
values at
different environmental urea concentrations ureaC
values for: (a)
cationic and (b) anionic urease-loaded hydrogels.
Fig. 3.21 Influence of the environmental sodium chloride concentration NaClC
on the distributive profile of the electrical potential at ureaC
= 10
mM for: (a) cationic and (b) anionic charged urease-loaded
hydrogels.
Fig. 4.1 The comparison between the present numerical result and the published
experimental observations: (a) the oxygen saturation with neonatal
hemoglobin as a function of ambient oxygen levels; (b) the pH-
induced swelling deformation of the hemoglobin-loaded hydrogel.
Fig. 4.2 The saturation of (a) newborn and (b) adult oxyhemoglobins as a
function of ambient oxygen level with environmental pH of 7.4 via
published experimental observation and present numberical result.
For newborn hemoglobin: kOxyHb/ kOxy= 170 mMs-1, and for adult
hemoglobin: kOxyHb/ kOxy= 50 mMs-1 .
Fig. 4.3 (a) A comparison between present numerical result and published
experimental observations for the pH-induced swelling deformation
behaviors of a typical polyampholyte hydrogel at environmental
temperature 25 °C. (b) The pH-induced swelling deformation
behaviors of hemoglobin-loaded polyampholyte hydrogel, while inset
of Fig.3(b) shows the pH-actuated osmotic pressure response of
hemoglobin-loaded hydrogel. The osmotic pressure response p of
the hydrogel is the differential osmotic pressure between the hydrogel
and the hydrogel at isoelectric point.
Fig. 4.4 (a) The impact of hemoglobin loading on the oxygen-induced swelling
deformation of the hemoglobin-loaded polyampholyte hydrogel. (b)
The influence of fixed acidic and basic groups densities on the pH-
driven reactive performances of the hemoglobin-loaded
polyampholyte hydrogel.
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Fig. 4.5 The impact of environmental pH on (a) newborn hemoglobin saturation
with oxygen and (b) hemoglobin-mediated oxygen activity in the
hydrogel as a function of ambient oxygen level.
Fig. 4.6 (a) The pH-induced swelling deformation behaviors and (b) the pH-
actuated osmotic pressure response of hemoglobin-enriched
polyampholyte hydrogel loaded with different strengths of fixed
acidic and basic groups at ambient oxygen O2 level of 160 mmHg.
The osmotic pressure response p of the hydrogel is the differential
osmotic pressure between the present hydrogel and the hydrogel at
isoelectric point.
Fig. 4.7 The pH-induced (a) net charge concentration and (b) electrical potential
response of hemoglobin-enriched polyampholyte hydrogel loaded
with different strengths of fixed acidic and basic groups at ambient
oxygen O2 level of 160 mmHg.
Fig. 4.8 The pH-actuated osmotic pressure response of hemoglobin-enriched
polyampholyte hydrogel loaded with different strengths of fixed
acidic and basic groups at ambient oxygen O2 level of 160 mmHg
for: (a) 0AC = 200 mM and
0BC = 100 mM and (b)
0AC = 100 mM
and 0BC = 200 mM. The osmotic pressure response
p of the hydrogel
is the differential osmotic pressure between the present hydrogel and
the hydrogel at isoelectric point. The insets of (a) and (b) are its
corresponding pH-actuated swelling deformation behaviors.
Fig. 4.9 The pH-actuated electrical potential of hemoglobin-enriched
polyampholyte hydrogel loaded with different strengths of fixed
acidic and basic groups at ambient oxygen O2 level of 160 mmHg
for: (a) 0AC = 200 mM and
0BC = 100 mM and (b)
0AC = 100 mM
and 0BC = 200 mM.
Fig. 4.10 Comparison of membrane swelling deformation as a function of
environmental salt concentration via present numerical simulation
and published experimental observation for: (a) polybasic and (b)
polyampholytic membranes.
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Fig. 4.11 Comparison of membrane surface conductivity as a function of
environmental salt concentration via present numerical simulation
and published experimental observation for AMX and CMX charged
polymeric system.
Fig. 4.12 The (a) swelling deformation, (b) oxygen loading, (c) counter-
concentration, (d) co-ion concentration, (e) electrical potential and (f)
surface conductivity of the hemoglobin-loaded polyacidic membrane
with initial fixed charge concentration of 100 mM as a function of
ambient oxygen level, when operating at environmental pH of 7.40
and temperature of 37 °C.
Fig. 4.13 The (a) swelling deformation, (b) oxygen loading, (c) counter-
concentration, (d) co-ion concentration, (e) electrical potential and (f)
surface conductivity of the hemoglobin-loaded polyacidic membrane
with initial fixed charge concentration of 100 mM as a function of
environmental salt concentration, when operating at environmental
pH of 7.40 and temperature of 37 °C.
Fig. 4.14 The effect of initial fixed charge concentration on (a) swelling
deformation, (b) oxygen loading, (c) counter- concentration, (d) co-
ion concentration, (e) electrical potential and (f) surface conductivity
of the hemoglobin-loaded polyacidic membrane with initial fixed
charge concentration of 100 mM as a function of environmental salt
concentration, when operating at environmental pH of 7.40 and
temperature of 37 °C.
Fig. 4.15 The (a) swelling deformation, (b) oxygen loading, (c) counter-
concentration, (d) co-ion concentration, (e) electrical potential and (f)
surface conductivity of the hemoglobin-loaded polyacidic, polybasic
and polyampholytic membranes with initial fixed charge
concentration of 100 mM as a function of environmental salt
concentration, when operating at environmental pH of 7.40 and
temperature of 37 °C.
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List of Tables
Table 3.1 Input data for the numerical simulation of the multiphysics model for
urease-loaded hydrogel.
Table 4.1 Input data for the numerical simulation of the multiphysics model for
hemoglobin-loaded hydrogel.
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Chapter 1 Introduction
In this chapter, the background and motivation, objective and scopes and the
layout of this thesis are presented.
1.1 Background and Motivation
Environmentally responsive hydrogels are the go-to choice of material in the
field of medicine for two main reasons: (1) low in toxicity and (2) excellent
biocompatibility [1]. It is well-established that the hydrogel exhibits a variety of
practical electromechanical properties which include super absorption capacity,
great swelling/de-swelling capability, high counter-ion permeability and
excellent catalytic behavior [2], leading to the exploitation of the hydrogel in
numerous bioengineering applications, such as surgical actuators, dialysis
membranes and drugs. One of the earliest application of the hydrogel in
medicine is current contact lens, as proposed by Wichterle and Lim [3],
replacing plastic-based first-generation contact lens which was usually
associated with high material toxicity. The environmentally responsive
hydrogels are usually functionalized with stimuli-reactive neutral/charged
monomers [4] which results in the hydrogel to exhibit coupled bio-chemo-
electro-mechanical responses, when subjected to environmental cues, including,
but not limited to, ion concentration, ionic strength, light intensity, pressure,
magnetic or ultrasound [5-7].
A major advancement in the research front of environmentally-responsive
hydrogels is the development of bio-responsive hydrogel where researchers
integrated bioactive agents into the hydrogel, which initiates a new branch of
(bio) analyte-specific environmental-responsive hydrogels. Thereby, the bio-
responsive hydrogel usually consists of two main functional components: (i)
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sensor and (ii) actuator. The immobilized bioactive agent in the hydrogel acts as
the biosensor component due to its highly selective chemical reactions. Once
the bio-responsive hydrogel encounters the relevant stimulus, it generates a
chemical reaction in the hydrogels. This triggers a mechanical response by the
polymeric network chains, which behave as the actuator component of the
system. Literature surveys revealed that the bio-responsive hydrogels include,
but not limited to, glucose oxidase-, urease-, lipase-, hemoglobin-, antigen- or
antibody-loaded hydrogels [2, 8-11]. These exciting new biomaterials show
tremendous application potentials in a broad range of bioengineering medical
devices which explains the emergence of new bio-responsive hydrogel-related
works in academic databases. Hence, it is worthwhile and necessary to examine
the fundamental mechanism of these bio-responsive hydrogels to address
paucity of information in the literature.
In the quest to investigate the newly developed material, scientists
employed costly and time-consuming trial-and-error experimental technique
conducted to investigate the behaviors of the hydrogels under varying
environmental conditions. As a result of the trial-and-error experiment,
researchers were often led to off-tangent research directions [4]. Interestingly,
multiphysics modeling is a viable alternative option for obtaining a further
understanding into the reactive behaviors of the bio-responsive hydrogel.
However, the published mathematical models in open literature are incapable of
predicting the coupled bio-chemo-electro-mechanical responses of the
hydrogels [10, 12]. It is also noted that the published mathematical models
failed to take into account the impacts of the environmental circumstances on
the performance of the bio-reactive hydrogels, especially pH, temperature or
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ionic strength. In order to overcome these shortcomings, it is worthwhile to
develop multiphysics models for the simulation of the bio-responsive hydrogels
for predicting the coupled multi-domain responses of the bio-responsive
hydrogels in respond to specific stimulus under varying environmental
conditions. The models will be able to elucidate the fundamental mechanism
and key parameters of the hydrogels and then to achieve an optimization of the
hydrogel-based machines, such as sensors/actuators and bio-micro-fluidic
valves [13]. Therefore, due to the huge diversity in applications of these bio-
responsive hydrogels, it is worthwhile and necessary to develop a detailed
theoretical model for describing and getting a greater understanding into the
responsive performance of the hydrogels.
1.2 Objective and Scope
The aim of this present research work is to develop two multiphysics models
for simulation of the behavior of urease- and hemoglobin-loaded hydrogels by
coupling the electrical, chemical, and mechanical fields together to obtain a
greater insight into such materials. The specific objectives are detailed as
follows.
❖ Model development. Two multiphysics models are developed to describe
the fundamental mechanism of urease-loaded and hemoglobin-incorporated
hydrogels by merging chemo-electro-mechanical domains. In the model, a
correlation between the functional group (urease/hemoglobin) and
environmental conditions is integrated into the model, capturing the
bioactivity of the system.
❖ Model validation. In order to develop a robust mathematical model, the
present numerical results obtained from the model are validated against the
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experimental observations, where good agreements are achieved for both
models.
❖ Model exploration. The model is numerically solved, where the present
numerical results are utilized to investigate qualitatively and quantitatively
the performance of the hydrogels in variation of biochemical, electrical, and
mechanical fields.
(i) Hydrogel performance in the biochemical field. The present work
improves the understanding of the bioactivity of fixed urease and
hemoglobin when operating in charged polymeric systems. In particular,
the effects of environmental conditions on chemical reaction of
urease/hemoglobin are also elucidated, including environmental solution
pH, ionic strength and temperature.
(ii) Hydrogel performance in the electrical field. The current work
describes the influence of environmental conditions on the electrical
performances of the hydrogels, especially its electrical potential acting
over hydrogel-solution interface. In addition, the effects of initial
immobile charge concentration are also examined on the hydrogel
electrical behaviors.
(iii) Hydrogel performance in the mechanical field. The present work
characterizes the coupled biochemical-electrical effects on the
responsive hydration-induced mechanical deformation of the hydrogel.
As such, the environmental-induced mechanical deformation is also
described when responding to the extreme of physiological conditions.
1.3 Outline
This thesis comprises of five chapters, as outlined below.
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Chapter 1 explains the motivation of investigating the fundamental
mechanism of urease-loaded and hemoglobin-enriched hydrogels followed by
the objective and scope and outline of the present thesis.
Chapter 2 examines the current literature on the experimental and
theoretical works on reactive performances of the bio-responsive hydrogels,
especially the urease-loaded and hemoglobin-incorporated hydrogels. In
addition, the modeling works are also examined to unveil the limitation of the
published mathematical model.
Chapter 3 formulates a multiphysics model to investigate the reactive
performance of urease-loaded hydrogel by incorporating bio-electrochemical
interactions between environmental urea-rich salt solution and functional
groups within the hydrogel. The model is directly validated by comparing
against published experimental results, where it reasonably captures the
environmental-induced reactive performances of urease-loaded hydrogel.
Chapter 4 describes a multiphysics model to examine the coupled
responses of hemoglobin-loaded polyelectrolyte hydrogel by accounting bio-
electrochemical interactions between environmental oxygen-rich salt solution
and functional groups within the hydrogel. The model is directly validated by
comparison with published experimental data, where it rationally characterizes
the dual oxygen-pH coupled stimuli behaviors of the hydrogel.
Chapter 5 entails the present theoretical efforts with recommendation for
future work.
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Chapter 2 Literature Review
This chapter deals with literature surveys of environmentally bio-responsive
hydrogels, especially protein-enriched hydrogels. After that, literature surveys
are also conducted for the modeling of hydrogel and protein equilibrium
responses.
2.1 Experimental Investigation on Bio-responsive Hydrogels
The word hydrogel is derived from the Latin prefix hydro where “hydro” means
water and “gel” referring to a solid structure [14]. As such, hydrogel is a three-
dimensional (3D) structure consisting of highly cross-linked polymeric network
chains [15-17], where the hydrogel is able to imbibe and retain large fluid
quantity in the system due to hydrophilic and elastic nature of the polymeric
network chains. Thereby, the polymeric system consists of large fraction of
water with a small polymer counterpart [3, 18], resulting in the hydrogel to
demonstrate mixed solid-liquid characteristics. The ingestion of fluid into the
hydrogel stretches its polymeric network chains, up till to a certain extent,
where the elastic-nature of the crosslinked polymeric network chains prevents
the dissolution of the polymeric system during hydration [19]. It is well-
established that the random-nature cross-linkage of the polymeric network
chains creates a porous structure, where the increase of hydrogel hydration
promotes the movement of mobile environmental cues into its system.
Interestingly, environmentally responsive hydrogel is capable of converting
environmental cues into an electromechanical response, when the polymeric
network chains encounter the environmental cues, such as glucose [20], urea
[21], salt [22], temperature [23], light [24] and pH [25]. Literature survey
revealed that the focus of investigations into environmentally responsive
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hydrogels is usually pointed to pH-responsive hydrogel, where it is synthesized
by incorporating weakly acidic or basic group. As such, the ionization state of
the immobile charge group can be tuned by manipulating the environmental pH,
where the increase of pH-governed ionized immobile charge component
concentration usually induces the hydration of the polymeric system. The
polymeric system usually contains anionic and/or cationic components [26-28].
The microscopic structure of an ionic hydrogel in which is functionalized
with immobile charged components, constructing the hydrogel. It is commonly
known that hydrogel is able to swell up to 1000-fold of its initial size, due to the
electrostatic interactions of fixed-fixed and fixed-mobile and mobile-mobile
charge components in the polymeric system [29, 30]. Interestingly, the
mechanical performance of the polymeric hydrogel can be tuned with the
manipulation of the electrostatic interactions of the immobile and mobile charge
components via changing the environmental solution salt concentration [31].
On the one hand, the anionic hydrogel consists of weakly acidic group on
the polymeric network chains, which forms cation mobile solutes and anionic
immobile charge group when subjected to environmental solution with specific
pH values, including poly(methacrylicacid)-, poly(acrylic-acid)-, poly(N-
isopropylacrylamide-co-acrylic-acid)- poly(methacrylic-acid-ethylene-glycol)-,
poly(methacrylic-acid-co-methacryloxyethyl-glucoside)-, and poly( N,N-
dimethylacrylamide-co-2-acrylamido-2-methylpropane sulfonic acid)-based
hydrogel [32-49]. It is known that the fixed acidic group of the hydrogel
remains un-ionized, when subjected to environmental pH < acid-base
dissociation constant pKa which makes the hydrogel to behave like a neutral
hydrogel [50]. On the other hand, the cationic hydrogel composes of weakly
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basic group attached on the polymeric network chains which ionizes into anion
mobile solutes and cationic immobile charge group in response to
environmental solution, where the weakly basic group, such as amine, are
ionized by accepting positive charge mobile ions from the environment solution
[51]. This includes poly (2-diethylaminoethyl-methacrylate)-, poly (ethylene
oxide)- and poly (ethyleneimine)-based cationic hydrogels. It is well-
established that fixed basic group tends to remain unionized, when it is
immersed in environmental pH > acid-base dissociation constant pKa of the
basic group. The ionization of fixed acidic/basic group of the hydrogel causes
the increase hydration-induced swelling deformation of the hydrogel [50], due
to greater electrostatic repulsion between fixed-fixed charges coupled with
larger imbalance of counter-ion concentration over the hydrogel-solution
interface [52].
We now ask: What is bio-responsive hydrogel? In order to answer this
question, literature surveys are conducted in which they reveal that the bio-
responsive hydrogel is usually developed by incorporating biological
components onto the acidic/basic-loaded crosslinked polymeric network chains
(environmentally responsive hydrogel) via physical entrapment or UV radiation
[53, 54]. This enhances the specificity of hydrogel, where it only exhibits a
response upon stimulation by certain environmental bio-cues [19]. Due to the
interesting characteristics of bio-responsive hydrogel, such as specific catalytic
behaviors, excellent ion-sorption ability and high swelling capability, it is thus
exhibits great potential as a material platform for developing soft actuator,
dialysis membrane, gas carrier and artificial muscle, consequently requiring
greater attention amongst research to address paucity of information of such
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26
materials in the literature, where this thesis focuses on the investigation of both
urease-loaded and hemoglobin-incorporated hydrogels.
2.1.1 Urease Immobilized Hydrogel.
It is well-established that urease is a naturally occurring group of enzymes
found in bacteria, fungi, algae, invertebrates and plant. A literature surveys
unveil that majority of the published investigation aimed on the urease
derivation from different bacteria and various plants. Thereby, the plant-based
ureases were usually obtained from jack bean [55], soybean [56] and pigeon pea
[57], while the bacterial-origin ureases were typically extracted from Bacillus
pasteurii [58, 59], Proteus mirabilis [60] and Providencia rettgeri [56]. To
improve the yield of the enzyme, urease was typically purified, increasing the
urease catalytic activity by either: (i) crystallization [61] or (ii) chromatographic
[62]. In order to synthesize the urea-sensitive urease-loaded hydrogel,
researchers employed several techniques for the immobilization of urease in the
hydrogel, such as: (i) covalent binding [63], (ii) adsorption [64], (iii)
encapsulation [65], and (iv) entrapment [66]. Since the hydrogel retains large
amount of water inside its polymeric network chains, which mimics the
biological tissues to some extent, the urease immobilization in the hydrogel
provides a suitable microenvironment for the operation of the urease, improving
the stabilization of urease towards environmentally induced denaturation such
that it is physically more robust to free urease in a free solution [67].
Bayramoglu and Arica [53, 68-71], Baysal et al., [72-74] and Chen and Chiu
[63, 75-77] made significant contribution into the experimental investigations
of urease immobilized hydrogels.
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27
Bayramoglu and Arica developed urease-loaded hydrogels via irradiation of
UV or gamma rays on a series of various co-polymers, such as NVP, NIPAM,
HEMA and PEG with urease [53, 68-71]. They reported that improvement of
pH- and temperature-induced urease catalytic activity, and the urease
immobilized in poly(N-isopropylacrylamide-co-poly(ethyleneglycol)-
methacrylate) hydrogel retained almost ~100.0 % of the enzymatic activity after
eight times of use [53], demonstrating consistency with experimental
observation presented by Krajewska et al. where the immobilized urease in
polysulphone membrane was almost independent of the number of reuse after
17 times of usage [78].
Furthermore, Chen and Chiu proposed a polymeric system-based catalytic
reactor, where anion-exchange membrane and urease-loaded hydrogel were
clamped together to divide feed and stripping solutions [63, 75-77]. The five-
compartment electro-dialyzer employing urease functionalized with polymeric
system to remove urea from aqueous solution. Besides that, Lee et al. [79] and
Moynihan et al. [80] also developed hydrogel-based electro-dialyzers with the
protected urease for the regeneration of the dialysate solution. Interestingly,
Baysal et al. integrated urease in PEG-based hydrogel, where it was then
encapsulated within the living red blood cells [72-74] as an injectable urea
dialyzer for transport of reactive urease in the vasculature system.
Over the past decades, there were great efforts to immobilize urease in
hydrogels for potential applications as bioreactor, biosensor and bioseparator
[63, 80-82], in which the hydrophilic polymeric network chains: (i) provided
support; (ii) improved stability towards variation of pH, temperature and ionic
strength; (iii) increased shelf life for the immobilized urease [83-87].
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28
Unfortunately, the fundamental understanding of urease-loaded charged
hydrogel [85] as a biofunctional membrane is still preliminary. Therefore, the
urease catalytic performance of the urease-loaded anionic and cationic hydrogel
is necessarily investigated, when subjected to variation of environmental cues,
such as pH, temperature, urea and sodium chloride.
A literature search reveals that the effect of environmental sodium chloride
concentration [88] still remains unclear on the performance of urease catalytic
in the charged hydrogel [89]. Usually the change in the environmental sodium
chloride concentration alters the fixed group density of the urease-loaded
charged hydrogel [90], forming a new microenvironment with different ionic
concentration and pH values in the hydrogel. This perhaps leads to the change
of urease catalytic activity due to the modification of the environmental sodium
chloride concentration. As such, the behavioral patterns of urease catalytic
activity in the anionic or cationic hydrogel are necessarily examined as a
function of the environment sodium chloride concentration. The literature
survey also reveals that, the previous investigations aimed only on the impact of
environmental pH and temperature on the catalytic activity in the charged
hydrogel [89, 91, 92]. Therefore, in-depth research on the urease catalytic
behaviors in the charged anionic or cationic hydrogel are necessarily elucidated
as a function environmental sodium chloride concentration to understand the
way sodium chloride modifies the reactive performances of the urease-loaded
hydrogels.
To our best knowledge, no study was involved in the urease catalytic
behaviors in the charged hydrogel with variation of environmental urea
concentrations at different catalytic and Michaelis constants. For a nonspecific
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29
immobilized enzyme in a charged polymeric environment, several interesting
experimental observations were reported on its catalytic behaviors [93],
including (1) the dilution of the enzymatic activity due to hydrophilic nature of
the polymeric system [50], (2) the ionization of the enzyme due to the acidic or
basic nature of the polymeric system [94], and (3) the alteration of optimum
operating conditions of the enzyme [94]. However, some experimental
observations seem to be conflicting each other, in which few of them
demonstrated that the enzyme activity improves when operating in a charged
polymeric environment [89], while another experiment showed that the
immobilization of enzyme into a charged hydrogel decreases its enzymatic
activity [95]. As such, it is worthwhile to develop a multiphysics model to
elucidate the urease catalytic activity in a charged hydrogel in response to urea
cue with variation of catalytic and Michaelis constants to systematically tailor
such materials.
2.1.2 Hemoglobin-loaded Hydrogel.
In general, free hemoglobin consists of two components, the organic and
inorganic parts [96-98], where the inorganic heme iron complex of hemoglobin
binds with oxygen O2 molecules [99-101], forming oxygen-heme iron
complexes [102]. Interestingly, the association of oxygen O2 with the iron
complex leads to a tense-to-relax structural change in the hemoglobin, altering
oxygen O2 affinity of the free hemoglobin [103-106]. In addition, formation of
the free oxygen-hemoglobin complex is also influenced by environmental pH,
where the affinity of hemoglobin towards oxygen O2 is decreased, when acidity
of the environmental solution increases. It is commonly known that despite the
attractive capability of the hemoglobin for sensing and capturing oxygen O2 in
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30
salt solution [107], a few crucial electrochemical reactions influence
performance of free hemoglobin in biological fluids including: (1) the
environmental pH-induced reduction of hemoglobin-oxygen activity [108] and
(2) the oxidization of oxygen-binding group (heme) in the hemoglobin [109].
Therefore, researchers began immobilizing hemoglobin into hydrogels as a
plausible strategy to enhance the performance of the hemoglobin for sensing
and storing of oxygen O2 in salt solution [108, 110-114]
Literature surveys unveil that there were tremendous investigation to realize
the possibility of integrating hemoglobin in the polymeric system, especially by
Chang et al. [115-117] and Palmer et al. [108, 112, 118, 119]. The focus of the
investigations of hemoglobin-hydrogel system were pointed to the: (i) pH-
induced swelling behaviors of hemoglobin-loaded hydrogel [2], (ii) hemoglobin
adsorption into hydrogel reported by Shirahama et al.[120], Vidal et al. [121],
Uysal et al. [122], and Dessy et al.[123], (iii) hemoglobin isolation from human
blood via hydrogel-based bio-selector, (iv) molecularly imprinted method for
developing hemoglobin immobilized hydrogels conducted by Xia et al. [124],
Uysal et al. [122] and Gou et al.[125].
In terms of application, there were great efforts to incorporate hemoglobin
into the hydrogels for developing bio-engineering devices, such as: (i)
biosensors by Sun et al.[126], Shan et al. [127], Chen and Lu [128], Zheng at
al. [129], Reddy et al. [130], and Zhong et al. [131] and (ii) oxygen carriers
reported by Philips et al. [132], Eike and Palmer [118], and Li et al. [133].
Unfortunately, literature reviews unveil that the reactive performances of
hemoglobin-loaded hydrogel remain poorly understood, in which no efforts
were conducted experimentally and theoretically to investigate the performance
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31
of hemoglobin-loaded polyampholyte hydrogel as a function of ambient oxygen
O2 coupled with environmental pH.
It is well known that the oxygen O2 and pH are two critical biomarkers in
the human system, which facilitate: (i) the detection of diseases, such as breast
and liver cancers [134] and (ii) the navigation of gas transport in hemoglobin-
rich blood [135]. As a result, it is important for an advanced biosensor to detect
and measure these two important biomarkers in biological fluids,
simultaneously. In addition, in terms of the hydrogel-based biofuel cell [136],
particularly the enzyme mediates pH-driven one [137], their performances are
usually limited by the oxygen O2 concentration in the microenvironment [138],
due to poor solubility of the physiological gas in the water-loaded polymeric
system. Therefore, it is necessary to incorporate the hemoglobin into the
polyampholyte hydrogel for developing and characterizing the oxygen-pH
stimuli coupled biosensing or oxygen-rich biofuel material platform [108], in
which the hemoglobin is inherently associated with the attractive oxygen O2
sensing and capturing properties [139], while the acidic-basic groups in the
hydrogel undergo ionization/deionization in response to environmental pH. In
other words, incorporation of both the hemoglobin and immobile charge
component with the polymeric network chains give rise to its dual oxygen- and
pH-reactive behaviors. Therefore, it is worthwhile and necessary to elucidate
the reactive performances of the hemoglobin-loaded hydrogel, especially via
theoretical modelling for systematically tailoring such materials as the
biosensing and biofuel cell platforms [140, 141].
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32
2.1.3 Other Bio-responsive Hydrogels
On the one hand, lipase enzyme can be integrated with the hydrogel for
catalyzing the esterification of lipid. It is commonly known in biology that lipid
is a group of organic groups where it is insoluble in water, including hormones,
fats and oils. Literature searches reveal that majority of the published
investigation aimed on the extraction of lipase from Candida rugose which is a
group of fungi or Pseudomonas aeruginosa which is a type of microbial [142-
147]. The focus of the investigations of lipase-loaded system was usually aim
to: (i) the stabilities of the immobilized lipase [148-151], (ii) the enhancement
of lipase catalytic activity by improving accessibility of the lipid into the
composite hydrogels [152-160], (iii) the lipase leaching and activity in lipase-
alginate/chitosan hydrogel as reported by Betigeri and Neau [161], (iv) the
coupled effects of pH and temperature on the immobilized lipase catalytic
performance [9, 162-165], (v) the mechanical performance of the hydrogelwhen
subjected to different environmental solvents [166, 167]. In terms of
application, the lipase-loaded hydrogels were usually explored as bioreactors
and super-absorbent [168, 169].
On the other hand, enzyme-sensitive biomaterial can be integrated with the
polymeric network chains for developing enzyme-induced biodegradable
polymeric systems. It is well-reported that the polymer biodegradation involves
the dissolution of its crosslinked polymeric network chains, which is facilitated
by enzymatic biological agents. Literature searches unveil that majority of the
published investigation aimed on the enzyme such as: (i) esterase, proteases,
collagenase and bovine serum albumin [170, 171]. In terms of application, the
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33
lipase-loaded hydrogels appear to be attractive in the field of medical,
bioreactor, and biomedical devices.
2.2 Modeling the Equilibrium of Hydrogel Responses and
Protein Activities
The continuum mathematical models in open-literature were developed for
capturing the environmentally responsive behavior of hydrogel and protein,
responding to different stimuli. Here, a good starting point for developing the
new mathematical model will be the mass, energy and momentum conservation
laws.
2.2.1 Transport Model
The transport model was firstly developed by De et al. [172] for investigating
the mechanical responses of the hydrogel, in which the model was extended by
later researchers [173-176]. The model was derived from the mass conservation
law, or so-called the diffusion equation, where the ionic transport is caused by
its concentration gradient over the hydrogel-solution interface. In order to
consider the migration of ion transport in the system, the model was extended to
include the electrical potential effect on the mobile ion concentration in the
hydrogel, coupling the effect of diffusion-governed with the migration-
administered ionic transport [177], given as follows
( )1
. 0 1,.2,3,...,k k k k k kk
Dc RT c z F v r k N
RT c
− + + = =
(2.1)
where kD , kc , and R T indicate the diffusivity tensor (m2/s), concentration
(mM) , universal gas constant (8.314 J/mol K) and absolute temperature (K).
, and kz F indicate the valence number, Faraday constant (96,487 C/mol) and
-
34
electrical potential (mV). kv represents coefficient of the reaction rate and kr
refers to the source term (mM/s).
The electrical potential is modelled by the Poisson equation, which
correlated the electrical potential acting over the hydrogel-solution with total
charge concentration of the system
2
10
N
k k f f
kr
Fz c z c
=
= − +
(2.2)
where r is the relative dielectric constant of the surrounding medium, 0 the
vacuum permittivity (8.85418 x 10-12 C2/N m2), fz valence of the fixed charge
concentration and fc density of the immobile charge groups within the
hydrogel, which is given as [178]
0
s
m Hf
a H
ccc
H K c
(2.3)
where 0s
mc the density of the total ionizable immobile charge groups within the
dry gel , H the hydration of the hydrogel and aK acid-base dissociation
constant of the immobile charge groups attached with the hydrogel.
The mechanical deformation problem for hydrogel was characterized using
the conservation of linear momentum law, which can be written as
0 = (2.4)
where is the stress tensor. The transport model suggested that the hydrogel
deformation was driven solely by imbalance of ionic distribution the hydrogel-
solution, given as
k k
k
p RT c c (2.5)
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35
where p is osmotic pressure acting over the interface of hydrogel-solution and
kc is concentration of external solution. The Poisson-Nernst-Planck equations
are integrated with the non-linear mechanical equation to complete model
formulation.
The second transport model found in the literature is termed the Stefan-
Maxwell model, or termed the multicomponent diffusion equation, it was
earlier-established by Bisschop et al. [179], and followed by other subsequent
researchers [178-182]. The model was derived based on the relationship
between driving-friction forces, given as
, ,dr k fr kF F (2.6)
The relationship was further re-termed into the following
1/3
2
0
2 11 1
pnp p p p s p
c p eff
dIn v
dr M Q Dv v
(2.7)
where the volume fraction of polymer is given as p , is Flory-Huggins
parameter, v is partial molar volume of the solvent n is density of the
polymer network in dry solid state, and cM is average molecular weight
between two junctions of the polymeric network chain. The network
functionality Q at a cross-link usually has a value of 3 or 4, depending on the
type of polymer and cross-linking agent. On the right side of the equation, effD
is effective diffusion coefficient, sv and pv are of solvent and polymer
permeation flow linear velocities.
As such, the model postulated that the hydration of the hydrogel was caused
by (i) the mixing of solvent and polymer, (ii) the interaction between hydrogel-
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36
environmental solution and (iii) the elastic forces exhibited by the hydrogel. It
is noted that, one major concern associated with this model is the increase effort
for determining the transport parameter i.e. the effective diffusion coefficients,
especially when dealing with multi-components ion systems.
2.2.2 Multiphasic Mixture Model
As the name suggested, the multiphasic mixture model assumed that the
hydrogel consisted of three phases namely, solid, water and ion, in which each
phase is associated with its governing equation. The degree of hydration-
induced deformation of the hydrogel is governed by electrochemical potential
gradient of water and mobile ions between the hydrogel-solution, in which the
deformation of the hydrogel is limited to (1) the elastic contractive forces
exerted by the hydrogel and (2) the frictional forces between the different
phases and solvent.
Interestingly, Feng et al. coupled the multiphase model with the transport
model, mentioned above [183], where the conservation law of momentum was
employed for each phases, given as
0
0
0
p p w f pw w p
w k wp p w wk k w
k
w k k k iw w i
k
p c F f
p RT c f f
RT c Fz c f
v v
v v v v
v v
(2.8)
where equation 2.8 can be simplified into
1 0p p (2.9)
and thus, the flow of water relative to polymer network can be written as
ww p k k
wp
p Fz cf
v v (2.10)
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37
The Nernst-Planck equation is used to describe the ionic and molecular
transports of the hydrogel, given as
( ) 1.
w k k wk k k
k
c Dc RT c z F
t RT c
= +
(2.11)
The hydration-induced deformation of the hydrogel is taken to be a small
deformation-like behavior, given as
tr 2p 1 (2.12)
where and are Lame coefficients of the polymeric network chains, and
is strain tensor of the hydrogel. The multiphasic model considered the multi-
body interaction of the different phases, which was ignored in the transport
model. However, this increases the complexity of the model with the
introduction of effective diffusion coefficients into the mathematical
formulation. On a side note, it is known that the molecular dynamics model
investigates the behavior of the hydrogels in microscopic scale. However, when
dealing millions of solutes movements between hydrogel-solution system, it is
often much more practical to work at a macroscopic level with continuum
models.
2.2.3 Urease Enzymatic Activity
Michaelis and Menten developed a general continuum mathematical model to
characterize the binding of enzyme with its substrate [184], where it describes
the enzyme kinetics as a function of its substrate concentration, given as
follows
catM kK
act act actE S ES E P (2.13)
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38
where actE is active enzyme, S is substrate of the enzyme, actES is the enzyme-
substrate complex, and P is product of the enzyme-substrate reaction. KM is the
Michaelis constant where it measures the affinity enzyme to the substrate, while
kcat is enzyme rate of reaction which it indicates how fast the enzyme perform
after the formation of the enzyme-subtract complex. Briggs and Haldane
established a relationship between the Michaelis constant and the rate of
equations, given as [185]
2
1
catM
k kK
k
(2.14)
where k1 and k2 are the forward and backward reactions rate for the formulation
of actES complex. The reaction velocity Vmax is given as the function substrate,
where the equilibrium dissociation constant for the actES complex is given as
follows
act
S
act
E SK
ES
(2.15)
and
0 act actE E ES (2.16)
where Equations 2.15 and 2.16 is coupled to give
1Total S
act
E K
ES S
(2.17)
Thus the reaction velocity is given as [177]
max
S
V SV
K S
(2.18)
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39
Unfortunately, the Michaelis-Menten equation fails to account for the
environmental-induced, especially environmental temperature. Enzyme starts to
denature at temperature above the physiological temperature where it changes
the enzyme conformation, in which the non-reversible denaturation process
reduced its catalytic activity [186]
The equilibrium model was proposed to account for the denaturation of
enzyme as a function of environmental temperature, given as [187-190]. For
this purpose, a three-state (active, inactive and denatured states) mathematical
model is formulated which describes the enzyme activity via
eq inactK k
act inactE E X (2.19)
where two folded forms of enzyme are used, one being catalytically active
denoted by actE and the other one is catalytically inactive but not denaturized
denoted by inactE and X being the denatured form of the enzyme. eqK is the
equilibrium constant and inactk is the enzyme denaturation rate constant.
The enzyme velocity is thus given as
0max exp .
1 1
inact eqcat
eq eq
k K tk EV
K K
(2.20)
It is reported by that the simulated data obtained from the equilibrium model is
able to capture the experimental data more accurately than the conventional
model [190].
2.2.4 Reaction of Hemoglobin with Ligands
It is well-established that the formation of oxyhemoglobin on the polymeric
network chains is governed by equilibrium reactions between hemoglobin and
oxygen O2, as shown below
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40
2
2 2 2Hb/HbH O O Hb / O HbH
KO+ ++
(2.21)
where Hb / HbH+ and 2O Hb / 2O HbH+ refer to reduced/protonated reduced
and oxygenated/protonated oxygenated hemoglobins, while 2O
K denotes
association constant of oxygen O2 from the hemoglobin.
The immobile hemoglobin consists of ionizable functional components,
where the ionization of the hemoglobin is determined by the equilibrium
electrochemical reactions for the binding/unbinding of hydrogen H+ ion from
the hydrogel
1Hb H HbH
K+ ++ , and
2
2 2O Hb H O HbH
K+ ++
(2.22)
where 1K and 2K represent association constants for reduced and oxygenated
hemoglobins, and they can be written as 1 HbH H HbK+ + =
and
2 2 2O HbH H O HbK+ + =
. We assume that these reactions are assumed
to be in equilibrium locally due to faster reaction rate, in comparison with
progression of hydrogel hydration.
The total hemoglobin concentration [Hb]T in the hydrogel is can written as
T
2 2Hb Hb O Hb HbH O HbH+ + = + + +
(2.23)
where it consists of reduced and protonated reduced hemoglobins, and
oxygenated and protonated oxygenated hemoglobins.
The saturation of hemoglobin S with the oxygen O2 is given as
2
T
HbOS
Hb=
(2.24)
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41
where the oxygenated hemoglobin concentration 2HbO can be described as
follows
( )2 2 2
2 22
HbO O Hb O HbH
O Hb 1 HOK K
+
+
= +
= +
(2.25)
while the total hemoglobin concentration T
Hb can be written as
( )( )
T2
1 2 22
Hb Hb HbH + HbO
Hb 1 H + O 1 HOK K K
+
+ +
= +
= + +
(2.26)
2.3 Remarks
The literature surveys unveil that previously published experimental works
remain insufficient to characterize the reactive performances of such materials
including:
For urease-loaded hydrogel:
• how does the immobilized urease concentration influence
reactive performances of the hydrogel?
• how does different polymer monomers used to construct urease-
loaded hydrogel influence its urea-sensitive responses?
• osmotic pressure of the urease-loaded hydrogel at different:
➢ environmental conditions.
➢ urease kinetic properties.
➢ initial immobile charge component density.
• urease catalytic behavior of the hydrogel as a function of:
➢ urea concentration
➢ salt concentration
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42
The hemoglobin-enriched hydrogel reactive behaviors as a function of:
• hemoglobin loading
• immobile charge component density
• ambient oyxgen level
• environmental pH value
• coupled environmental salt- and oxygen-stimuli concentration
Therefore, these studies will be conducted and discussed in Chapter 3 and
4.
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43
Chapter 3 Multiphysics Model Development
to Investigate the Reactive Characteristics of
Urease-loaded Hydrogel
3.1 Introduction
This chapter deals with the multiphysics model development for investigating
the coupled bio-electro-chemo-mechanical reactive performances of urea-
sensitive hydrogel, incorporating the multi-physical interactions between the
immobilized urease and the environmental urea-rich salt solution. The model
includes Poisson-Nernst-Planck (PNP) equation for the transport of mobile
species between its microenvironment and the environmental solution coupled
with a nonlinear mechanical equation to account for the conversion of
biochemical and electric energies into its mechanical counterpart. In order to
capture the urea-induced behaviors of the hydrogel, two correlations are also
integrated into the multiphysics model to establish the relationship: (i) between
the urease reaction rate and diffusive concentration of urea, and also (ii)
between the fixed charge group density and diffusive hydrogen ionic
concentration. As such, the original contribution of Chapter 3 is that a novel
reaction rate equation is proposed to characterize the urease ionization and
denaturation states, capturing the urease activity as a function of pH coupled
with temperature. The model is directly validated by comparison with published
experimental data. This chapter is organized as follows. After the introduction
presented in Section 3.1, the multiphysics model is developed in Section 3.2,
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44
followed by Section 3.3 and 3.4 for validation and results discussions,
respectively. Finally, several remarks are drawn in Section 3.5.
3.2 Model Formulation
For developing the model, five items are necessarily described mathematically
including: (a) the mass transport between the hydrogel polymeric system and
environmental solution; (b) the urease-induced urea reaction in the polymeric
system; (c) the interplay of mobile-immobile charges in the polymeric system;
(d) the mechanical pressure arising from the urea-urease biochemical
interaction and (f) the volumetric behaviors of the polymeric system, as
visualized in Fig. 3.1.
The multiphysics model is developed with below premises:
(i) The hydrogel operates at constant temperature, where the urea hydrolysis
rate is assumed a constant in the polymeric system.
(ii) The enzyme is distributed and decorated homogenously on the polymeric
network chains.
(iii) The optimal temperature Top of the enzyme is assumed the midpoint
temperature of its transition between active and inactive forms Teq, namely
cateq GH 2 , in which eqH is the enthalpy change due to the transition
of active enzyme into its inactive counterpart, and catG is the enzymatic
reaction activation energy [204].
(iv) The polymeric system is associated with a macroporous nature, where the
mass diffusivity coefficients Dk remain the same, in either the polymeric
system or the environmental solution.
(v) The environmental solution is assumed unstirred.
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45
3.2.1 Conservation of Mass
In the model, the Nernst-Planck equation is employed for characterizing the
mass transport in the system, as shown below [205]
( )1 4 3, , , , ,kk k k k k
z FD C C v r k H OH NH HCO cation anion
RT
− + − + − + = =
C (3.1)
where 1−C the inverse of the right Cauchy-Green tensor and T−−− = FFC 1 1 . kD ,
kC , kkk rvz and, refer to diffusivity coefficient (m2/s), mass concentration (mM),
charge number, chemical reaction stoichiometric coefficient, and catalytic
reaction rate. and,,, FTR denote the universal gas constant (8.314 J/mol K),
environmental temperature (K), Faraday constant (96,487 C/mol), and electrical
potential of the polymeric system.
3.2.2 Conservation of Momentum
The hydrolysis of urea reaction in the polymeric system effects the reactive
swelling behaviors of the system due to changes of ionic concentration in the
hydrogel. It is well-established that the polymeric system achieves the
equilibrium swelling state, only if the osmotic swelling forces are balanced by
the restrictive forces display by the polymeric network chains. The momentum
conservation law can be utilized to describe the equilibrium swelling state
0P = (3.2)
given that P is the Piola stress.
3.2.3 Free Energy Imbalance Inequality
For formulating the mechanical constitutive equations for the polymeric system
at constant environmental temperature, the local free energy imbalance
inequality is used, given as follows [206]
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46
( ) 0uuu- 321W ++ (3.3)
where W is temporal Helmholtz free energy density of the polymeric system.
On the other hand, 321 uandu,u refer to the work per volume rate
incorporated into the polymeric system through the mechanical, electric and
chemical domains.
The free energy of the polymeric system are described by summing: [207]:
(i) the mechanical stretching free energy density of the elastic-nature polymeric
network chains W1, (ii) the mixing free energy density due to polymeric
network chains- environmental solution interaction W2, (iii) the mixing free
energy density due mobile charges-environmental solution interaction W3, and
(iv) the electric field free energy density due to mobile-immobile ions
interactions W4
4321 WWWWW +++= (3.4)
The elastic hydrogel stretching of free energy density, is given as [208]
( ) ( ) 3detn12tr2
1W1 −−= FFF
T
BTNk (3.5)
where TNkB is the ground-state shear modulus G, N is the polymeric chains
per volume of the dry polymer number (2.43 1025 m-3), and Bk is the
Boltzmann constant (1.38110-23 J/ K). By employing the method suggested
by Chester et al., the right Cauchy-Green tensor is written as ( ) es CC2
= [209],
where s is the swelling stretch and eC refers to the elastic right Cauchy-Green
tensor. Thereby, W1 is rewritten as
( ) ( ) ( ) 3detn12tr
2
1W
2
1 −−= FCesBTNk (3.6)
-
47
The mixing free energy density of hydrogel-environmental solution is given
as [209, 210]
( ) ( )
+++−=
CCCRTC
1
χ11n1nW2
(3.7)
where C is the number of water molecules in the environmental solution, is
the volume of a mole of fluid molecules and χ is the polymer-solvent
interaction parameter.
The mixing free energy density of environmental solution-mobile charges,
written as [211]
( ) −−=k
kk CCCRT 11n1nW3 (3.8)
The free energy density of the electric field formed due to mobile-immobile
charges interaction can be written as [208]
( )HCH .
2
1W
0
4 =Jr
(3.9)
where ( )3sJ = , H is the dielectric displacement, and 0 r is the relative
dielectric constant. The hydrogel volumetric change is caused the hydration-
induced swelling, where the W4 can rewritten [209]
( )( )HCH
+= e
r C3/1
0
412
1W
(3.10)
The mechanical power per volume can be written as [212, 213]
L:u1 J= (3.11)
where is the symmetric Cauchy stress tensor and L is the velocity, and1u can
be expressed as [209, 213]
( ) pJJp seTeTs −+= − FFPF :u1 (3.12)
-
48
where sF denotes the deformation gradient for the hydrogel swelling, while eF
denotes the deformation gradient of the hydrogel elastic deformation.
The rate of work done in electric field per volume can be expressed as [206]
k
k
kCzF −= HE2u (3.13)
The rate of work per volume caused by the movement of solution and
mobile solutes is written as [214]
+−
+−=
k
kk
k
k JJJJ k3u (3.14)
in which and k refer to chemical potentials of environmental solution and
mobile charge and and kJ J denotes the mass fluxes of environmental solution
and mobile charge. As such, the free energy imbalance inequality equation to
describe the reactive mechanical behaviors of the urease-loaded hydrogels is
written as
( )( ) ( )( ) ( ) ( )
( )( )
( ) ( )
( )( )
4/3
1/3
1/3 2/3 2
4/30
1 11 1 :
1
1 1 1tr 1n
3 1 11 1 1
11n
6 1
T TB e e e e e
B e
ke k
k r
Nk T C C pC
C χC Nk T RT
C CC C C
CC p RT C RT
C C
− − + + − + + +
+ + − + + +
+ + + + +
− − + + + +
F F H H F F F
C
H C H
0
kk k
k k
k
Cz F
C
+ −
+ +
J J
3.2.4 Biochemistry for Constitutive Equation
For utilizing equation 3.1, the conventional reaction rate kr which describes the
urease-mediated urea hydrolysis is expressed as [215]
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49
M
ureaE
catkK
CCkr act=
(3.16)
where actE
C and ureaC are the active urease and urea concentrations, respectively.
catk is the constant enzyme reaction rate and MK is the Michaelis constant. It is
well-established that the equation 3.16 neglected the environmental impacts on
the urease activity, especially environmental pH and temperature conditions.
Thereby, equation 3.16 can rewritten as follow for considering the effect of
temperature on urease catalytic behaviors [190]
EQMurea
EQ
ureakKKC
KCVr
+
+=
1max
(3.17)
in which maxV is the urease maximum reaction velocity, and EQ
Ecat
K
CkV act
+=
1max
[216],
−
=
TTR
HK
eq
eq
EQ
11exp is the equilibrium constant between active
and inactive urease, and 0E
C is the initial urease concentration.
The pH-induced urease denaturation is described mathematically by [217]
X
k
EOH
K
OHECCCC
inact
inact
EOH
act→+ −−(a)
(3.18)
X
k
EH
K
HECCCC
inact
inact
EH
act→+ ++(b)
(3.19)
where EH inact
C + and EOH inactC − are the ionized ureases concentrations and XC
refers to the denatured urease concentration [217, 218]. EOHK and EHK are the
equilibrium constant for the urease hydroxylation and protonation, and WK is
the water dissociation constant. Thereby, the initial and denatured urease
concentrations difference can be expressed as
-
50
++=−
+
+ EH
H
HEOH
WEXE
K
C
CK
KCCC
act1
0
(3.20)
Last, the chemical reaction rate kr for including the coupled temperature-pH
impacts on urease catalytic activity can be written as
++
+=
+
+
H
EH
W
HEOH
EQ
M
urea
M
ureak
C
K
K
CKK
K
C
K
CVr 1max
(3.21)
3.2.5 Electrical Field for Constitutive Equation
To incorporate the coupled impact of the electrical potential of polymeric
system and mobile charges transport in the model, the Poisson equation is
utilized [205]
( )
+−= −
k
ffkkr CzCzFJ1
0 C (3.22)
in which immobile ion concentration fC of anionic or cationic hydrogel [178]
0 0
or f fa H
f f
a aH H
CC CKC C
H K C H K C
+
+ +
= =+ +
(3.23)
where 0fC is the initial immobile charges in the hydrogel, H is the ratio of final
to the initial volume of the hydrogel, and Ka is the acid-base ionization constant
of the immobile ionizable components in the polymeric system.
3.2.6 Mechanical Field for Constitutive Equation
By employing equation 3.15, at equilibrium, the first bracket is equal to 0, in
which Piola stress P can be written as [208]
( ) ( )
( ) ( )
2/3 1/3
2/3
1 1
11
T
e B e
e
C p Nk T C
C
−
−
= − + + +
+ +
P F F
F H H
(3.24)
-
51
where the Piola stress P is summation of hydrostatic pressure p, elastic
polymeric network chains stress, and Maxwell stress. The Maxwell stress is
neglectable at mechanical equilibrium [206] , and we can write
( ) ( ) esBT
es TNkp FFP +−=−2 (3.25)
Again from equation 3.15, at steady-state, the chemical potentials of the
environmental and microenvironmental solutions are [219]
( ) ( ) ( )
( )( )
( )
2
1/3 2/3
11n
1 1 1
1 1tr
31 1
k
k
B e
CC χ
RT RTC C C C
Nk T pC C
= − + + + + +
− − −
+ +
C
(3.26)
C
C
RT kk
=
(3.27)
At Donnan equilibrium, the chemical potentials in the environmental and
microenvironmental solutions matches at the hydrogel-solution interface,
consequently the hydrostatic pressure p is expressed as [207]
( ) ( ) ( )
( )( )
( )
2
1/3 2/3
11n
1 1 1 1
1 1 1tr
31 1
kk
k
B e
C RT C χp RT C
J C C C
Nk TC C
= − − + + − + + +
− −
+ +
C
(3.28)
As seen in equation 3.28, the mechanical pressure on the interface between
hydrogel-solution consists of a few multi-physical components, including
polymer-solution, polymer-polymer, and mobile-immobile ions synergistic [220,
221].
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52
3.2.7 Boundary Conditions
In this section, the steady-state simulations are conducted, where the responses
of cylindrical urease-loaded hydrogels are examined one dimensionally. The
mechanical swelling of the hydrogels is assumed at radial direction, where
center (X = 0) of the hydrogel is under Neumann boundary conditions, whereas
the solution is subjected to Dirichlet boundary conditions [222].
For continuity, the Neumann boundary condition is assumed at the
cylindrical hydrogel (X = 0), given as follows
( ) 0at,...,,0,0 ===
=
−+ XNOHHkXX
Ck (3.29)
whereas the Dirichlet boundary conditions are applied at the environment
solution
( )NOHHkCC kk ,...,,0, −+=== (3.30)
Furthermore, equation 3.25 can be rewritten as
( ) ( ) ( )
++−=
X
uTNkp esBs 11
2P
(3.31)
in which the stretch s is given as
o
gel
gelsss
L
L
X
u
X
x=
+=
= 1
(3.32)
where us is the swelling displacement and Lgel is the hydrogel radius.
By equation 3.32, the Piola stress P is again recast
( )
++
+−=
X
uTNkp
X
u sB
s 111
2
P (3.33)
and by substituting equation 3.33 into the momentum equation 3.2 we obtain
011
2
=
++
+−
X
uTNkp
X
u
X
sB
s (3.34)
-
53
Last, the swelling stretch s acting over the hydrogel-solution interface
gel
Bs LX
p
TNk== ,
(3.35)
For simulating the reactive performance of the hydrogel, COMSOL
Multiphysics 5.1 software is used to execute the multiphysics model, due to its
ability to solve coupled governing and constitutive equations. First, the
immobile ion density equation 3.23 is solved for getting the immobile charge
concentration fC associating to applied input parameter and conditions. Next,
the Nernst-Planck equation 3.1 which is coupled to the Poisson equation 3.22
are computed numerically to obtain the converged solution of mobile mass
concentration kC and electric potential . Then, the kC is inserted inside
equation 3.25 to obtain the S . Lastly, above mentioned steps are repeated till
all independent variables achieve convergence, as visualized in Fig. 3.2.
3.3 Validation of Model
To probe the accuracy of the model to capture the responsive characteristics of
urease-loaded hydrogel, the current numerical finding is necessarily examined
with published experimental result, especially for: (i) the mechanical behaviors
of the hydrogel and (ii) the biochemical activities of the hydrogel under varying
environmental conditions.
3.3.1 Mechanical Behaviors of the Hydrogel
For examining the multiphysics model, the equilibrium urea-mediated
mechanical behaviors of the polymeric system are investigated against the
published experimental observation. In the experimental study performed by
Ogawa and Kokufuta [10], the urease-loaded p(NIPAM)-based polymeric
system is submersed in maleate-buffer of 5 mM, consisting urea concentration
-
54
of 1 mM, with pH = 4 and temperature of 35 oC, where urease concentration
0EC of 0.2 to 8.0 mg/mL is functionalized in the hydrogel. The input needed to
execute the model are: μm2500=L , 1kz = (cationic hydrogel), 0
fC =0.775
mmol/g, 5.5710aK−= , and the rest are tabulated in Table 3.1.
Fig. 3.3 demonstrates that the swelling of the hydrogel is invariant at
concentration of immobilized urease 0E
C larger than 0.5 mg/mL, where the
immobile enzyme strengthens the mechanical behaviors of the hydrogel,
limiting the swelling performance of the hydrogel. For hydrogel loaded with
smaller urease concentration 0E
C (e.g. < 0.5 mg/mL), the current numerical
results overpredicted the experimental observations. Fortunately, the difference
between numerical and experimental results are still within the experimental
error of 13 m , as visualized in Fig. 3.3.
Apart from that, Ogawa and Kokufuta also examine the impact of hydrogel
initial radius on its hydration-induced swelling deformation as function of
the immobile urease concentration, as visualized in Fig. 3.4 [10]. Therefore,
the current multiphysics model is re-performed, revealing that the increase of
the urease enzymatic activity in the hydrogel enhances the deionization of
the cationic immobile charge group. This depletes the ionic concentration
difference between the hydrogel and its environmental solution, which leads
to swelling-to-collapse volumetric behavior of the hydrogel. In addition, the
swelling of the hydrogel seems to be almost independent of the urease
concentration above 1.0 mg/mL. This is perhaps due to the limiting
concentration of urea in the hydrogel, which explains the invariant swelling
deformation of the hydrogel. It is noted that from Fig. 3.4, the current
numerical results usually underestimate the experimental equilibrium
swelling behaviour of the hydrogel as a function of urease concentration.
-
55
Fortunately, the errors of the present numerical simulations are usually
within the reported experimental error of 13 m , where it can be concluded
that the numerical result agrees well with the experimental findings. Hence,
as seen in Figs 3.3 and 3.4, the present numerical results seem to agree well
with the published experimental data, achieving a difference of ~3.0 %.
It is commonly known that the volumet